• Nonlinear Functional Analysis and Applications
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• v.27 no.4
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• pp.797-805
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• 2022

Let P(z) be a polynomial of degree n. A well-known inequality due to S. Bernstein states that if P ∈ P_n, then $$\max_{{\mid}z{\mid}=1}\,{\mid}P^{\prime}(z){\mid}\,{\leq}n\,\max_{{\mid}z{\mid}=1}\,{\mid}P(z){\mid}$$. In this paper, we establish some extensions and refinements of the above inequality to polar derivative and some other well-known inequalities concerning the polynomials and their ordinary derivatives.

• ALGAE
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• v.37 no.2
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• pp.175-183
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• 2022

Polar microorganisms produce physiologically active substances to adapt to harsh environments, and these substances can be used as biomedical compounds. The green microalga Micractinium variabile KSF0031, which was isolated from Antarctica, produced phytol, a natural antimicrobial agent. Furthermore, several polyunsaturated fatty acids (PUFAs), including omega-3, exhibit antioxidant properties. Here statistical methods (Plackett-Burman design and Box-Behnken design) were used to optimize the culture medium of KSF0031 to improve biomass production, and K₂HPO₄, MgSO₄·7H ₂O, and ammonium ferric citrate green (AFCg) were selected as significant components of the culture medium. Changes in the concentration of K₂HPO₄ and MgSO₄·7H ₂O as positive factors and AFCg as a negative factor affected cell growth to a remarkable degree. The biomass production in a 100 L culture using the optimized medium for 24 d at 18℃ was improved by 37.5% compared to that obtained using the original BG-11 medium. The quantities of PUFAs and phytol obtained were 13 mg g^-1 dry cell weight (DCW) and 10.98 mg g^-1 DCW, which represent improved yields of 11.70% and 48.78%, respectively. The results of this study could contribute to an improved production of phytol and fatty acids from Antarctic microalgae in the biomedical industry.

• Nonlinear Functional Analysis and Applications
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• v.27 no.1
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• pp.141-148
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• 2022

Let p(z) be a polynomial of degree n having no zero in |z| < k, k ≤ 1, then Govil proved $$\max_{{\mid}z{\mid}=1}{\mid}p^{\prime}(z){\mid}{\leq}{\frac{n}{1+k^n}}\max_{{\mid}z{\mid}=1}{\mid}p(z){\mid}$$, provided |p'(z)| and |q'(z)| attain their maximal at the same point on the circle |z| = 1, where $$q(z)=z^n{\overline{p(\frac{1}{\overline{z}})}}$$. In this paper, we extend the above inequality to polar derivative of a polynomial. Further, we also prove an improved version of above inequality into polar derivative.

• Nonlinear Functional Analysis and Applications
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• v.28 no.3
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• pp.731-751
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• 2023

If p(z) is a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1, then for any complex number α with |α| ≥ k, and r ≥ 1, Aziz [1] proved $$\left{{\int}_{0}^{2{\pi}}\,{\left|1+k^ne^{i{\theta}}\right|^r}\,d{\theta}\right}^{\frac{1}{r}}\;{\max\limits_{{\mid}z{\mid}=1}}\,{\mid}p^{\prime}(z){\mid}\,{\geq}\,n\,\left{{\int}_{0}^{2{\pi}}\,{\left|p(e^{i{\theta}})\right|^r\,d{\theta}\right}^{\frac{1}{r}}.$$ In this paper, we obtain an improved extension of the above inequality into polar derivative. Further, we also extend an inequality on polar derivative recently proved by Rather et al. [20] into L^r-norm. Our results not only extend some known polynomial inequalities, but also reduce to some interesting results as particular cases.

• Korean Journal of Mathematics
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• v.29 no.3
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• pp.631-638
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• 2021

In this paper, some sharp inequalities for ordinary derivative P'(z) and polar derivative D_αP(z) = nP(z) + (α - z)P'(z) are obtained by including some of the coefficients and modulus of each individual zero of a polynomial P(z) of degree n not vanishing in the region |z| > k, k ≥ 1. Our results also improve the bounds of Turán's and Aziz's inequalities.

• KOREAN JOURNAL OF PACKAGING SCIENCE & TECHNOLOGY
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• v.10 no.1
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• pp.7-14
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• 2004

Migration behavior of selected solvents and food samples showed differences of the chemical structures and polarities, the food samples which have similar polar expresses more higher affinity than different polar degrees. Water which is polar has a highest partitioning coefficient values on polar isopropanol, and oil which is nonpolar has highest partitioning value on non-polar toluene. The increasing order of partitioning values was accord with increasing water contents in food samples. It is showed that the wheat flour with 13.2% moisture content has the highest partitioning coefficient values on the isopropanol with -OH. Kp value of sugar showed remarkable lower partitioning coefficient values than other food samples due to high degree crystallinity. This phenomenon can be predictable with ${\delta}$ values, because order of partitioning coefficient values which comes out through the experiment and the sequence of Hildebrand solubility parameter value difference between food sample and printing ink solvent correspond almost. This Hildebrand solubility parameter value can be easily applied to the food package industry because the effect of food-safety can be considered without passing through complicated steps by using this method.

• Nonlinear Functional Analysis and Applications
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• v.28 no.4
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• pp.943-956
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• 2023

Let p(z) be a polynomial of degree n and for any complex number 𝛽, let D_𝛽p(z) = np(z) + (𝛽 - z)p'(z) denote the polar derivative of the polynomial with respect to 𝛽. In this paper, we consider the class of polynomial $$p(z)=(z-z_0)^s \left(a_0+\sum\limits_{{\nu}=0}^{n-s}a_{\nu}z^{\nu}\right)$$ of degree n having a zero of order s at z₀, |z₀| < 1 and the remaining n - s zeros are outside |z| < k, k ≥ 1 and establish upper bound estimates for the maximum of |D_𝛽p(z)| as well as |p(Rz) - p(rz)|, R ≥ r ≥ 1 on the unit disk.

• Ocean and Polar Research
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• v.30 no.4
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• pp.407-418
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• 2008

Seawater samples were collected at discrete depths from five stations across the polar front in the Drake Passage (Antarctic Ocean) by the $20^{th}$ Korea Antarctic Research Program in December, 2006. Nitrate concentrations of seawater increase with depth within the photic zone above the depth of Upper Circumpolar Deep Water (UCDW). In contrast, ${\delta}^{15}N$ values of seawater nitrate decrease with depth, showing a mirror image to the nitrate variation. Such a distinct vertical variation is mainly attributed to the degree of nitrate assimilation by phytoplankton as well as organic matter degradation of sinking particles within the surface layer. The preferential $^{14}{NO_3}^-$ assimilation by the phytoplankton causes $^{15}{NO_3}^-$ concentration to become high in a closedsystem surface-water environment during the primary production, whereas more $^{14}{NO_3}^-$ is added to the seawater during the degradation of sinking organic particles. The water-mass mixing seems to play an important role in the alteration of ${\delta}^{15}N$ values in the deep layer below the UCDW. Across the polar front, nitrate concentrations of surface seawater decrease and corresponding ${\delta}^{15}N$ values increase northward, which is likely due to the degree of nitrate utilization during the primary production. Based on the Rayleigh model, the calculated ${\varepsilon}$ (isotope effect of nitrate uptake) values between 4.0%o and 5.8%o were validated by the previously reported data, although the preformed ${\delta}^{15}{{NO_3}^-}_{initial}$ value of UCDW is important in the calculation of ${\varepsilon}$ values.

• Nonlinear Functional Analysis and Applications
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• v.29 no.2
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• pp.451-460
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• 2024

If a₀ + Σⁿ_ν=μ a_νz^ν, 1 ≤ µ ≤ n, is a polynomial of degree n having no zeroin |z| < k, k ≥ 1 and p'(z) its derivative, then Qazi [19] proved $$\max_{{\left|z\right|=1}}\left|p\prime(z)\right|\leq{n}\frac{1+\frac{{\mu}}{n}\left|\frac{a_{\mu}}{a_0} \right|k^{{\mu}+1}}{1+k^{{\mu}+1}+\frac{{\mu}}{n}\left|\frac{a_{\mu}}{a_0} \right|(k^{{\mu}+1}+k^{2{\mu}})}\max_{{\left|z\right|=1}}\left|p(z)\right|$$ In this paper, we not only obtain the L^r version of the polar derivative of the above inequality for r > 0, but also obtain an improved L^r extension in polar derivative.

• Tunnel and Underground Space
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• v.20 no.4
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• pp.241-251
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• 2010

The polar region is in the limelight for an academic worth as well as plenty of natural resources. The study on the polar region was reviewed for better understanding of the polar region and its rock properties. The antarctica has a windy and dry climate along with the lowest temperature on the earth. The thermal distribution according to depth in the area was reported: The freezing-thawing process was repeated in shallow depth, and the temperature falls down below zero under the specific depth. There is a great temperature difference between the atmosphere and rock. A research reported for the degree of weathering of the antarctic slope by using Schmidt hammer and Taffoni test. The rock specimens weathered by repeated freezing-thawing process were tested of the shore hardness and uniaxial compressive strength: The rock strength gradually decreased as the freezing-thawing process was repeated. The comprehensive mechanical properties of the polar region rocks and the relationship between the laboratory weathering test result and the real rock property change in the site remain as future research topics.